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arXiv:quant-ph/0506185 · Physical Review A

Quantum-limited velocity readout and quantum feedback cooling of a trapped ion via electromagnetically induced transparency

Peter Rabl, Viktor Steixner, P. Zoller

Published Oct 27, 2005• 18 citations
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Abstract

We discuss continuous observation of the momentum of a single atom by employing the high velocity sensitivity of the index of refraction in a driven $\ensuremath{\Lambda}$-system based on electromagnetically induced transparency. In the ideal limit of unit collection efficiency this provides a quantum-limited measurement with minimal backaction on the atomic motion. A feedback loop, which drives the atom with a force proportional to measured signal, provides a cooling mechanism for the atomic motion. We derive the master equation which describes the feedback cooling and show that in the Lamb-Dicke limit the steady state energies are close to the ground state, limited only by the photon collection efficiency. Outside of the Lamb-Dicke regime the predicted temperatures are well below the Doppler limit.

DOI
10.1103/physreva.72.043823

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We discuss continuous observation of the momentum of a single atom by employing the high velocity sensitivity of the index of refraction in a driven \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-system based on electromagnetically induced transparency. In the ideal limit of unit collection efficiency this provides a quantum-limited measurement with minimal backaction on the atomic motion. A feedback loop, which drives the atom with a force proportional to measured signal, provides a cooling mechanism for the atomic motion. We derive the master equation which describes the feedback cooling and show that in the Lamb-Dicke limit the steady state energies are close to the ground state, limited only by the photon collection efficiency. Outside of the Lamb-Dicke regime the predicted temperatures are well below the Doppler limit.
Summary form only given. The preparation of quantum systems in pure states is a prerequisite for coherent control. In this context laser cooling of trapped particles to the ground state of the confining potential is a powerful tool for preparing e.g. ions for quantum logic. In particular sideband cooling has been demonstrated to be the most efficient technique, allowing to achieve more than 99% of occupation of the ground state of the trap. This cooling technique relies on two conditions: the Lamb-Dicke regime, and the strong confinement regime. This latter condition, which allows the spectral resolution of individual sidebands of the motion, is however a limitation on the cooling time. We propose a model for cooling single ions inside a trap as an alternative to sideband cooling. The scheme uses electromagnetically induced transparency (EIT) in a /spl Lambda/-level scheme, where one of the transitions is dressed by a strong laser field. For detuning of the dressing laser, the absorption spectrum for the transition shows EIT when the cooling (probe) laser has detuning. Then, by tuning the cooling laser on the dark resonance (or EIT), for a suitable set of parameters and in the Lamb-Dicke regime the atoms are cooled to the ground state by the sideband transitions which are resonantly excited. The scheme achieves efficient ground state cooling and fast cooling rates using a continuous driving, instead of a pulsed one as in Raman sideband cooling schemes. An example is illustrated, where the average vibrational number is plotted as a function of the cooling time, and the final probability of occupation is plotted as a function of the vibrational state. This method may also be applied to reduce phase errors in coherent manipulations of trapped ions for quantum logic.
We investigate the cooling of trapped atoms by electromagnetically induced transparency under conditions of weak confinement and beyond the Lamb-Dicke limit, i.e., the spontaneous decay width is large compared to the trap oscillation frequency and the recoil energy is a substantial fraction of the vibrational energy spacing of the trap. Numerical solutions of the Liouville equation for a density matrix describing states of vibrational and electronic degrees of freedom show that vibrational cooling is feasible at even substantial values of the Lamb-Dicke parameter and under conditions of weak confinement, a situation where sideband pumping is inefficient. Our approach permits us to predict cooling efficiency and cooling rates under realistic experimental conditions for neutral atoms in optical dipole traps.
We propose a cooling scheme for a trapped atom using the phenomenon of cavity-induced double electromagnetically induced transparency (EIT), where the atom comprising of four levels in tripod configuration is confined inside a high-finesse optical cavity. By exploiting one cavity-induced EIT, which involves one cavity photon and two laser photons, carrier transition can be eliminated due to the quantum destructive interference of excitation paths. Heating process originated from blue-sideband transition mediated by cavity field can also be prohibited due to the destructive quantum interference with the additional transition between the additional ground state and the excited state. As a consequence, the trapped atom can be cooled to the motional ground state in the leading order of the Lamb-Dicke parameters. In addition, the cooling rate is of the same order of magnitude as that obtained in the cavity-induced single EIT scheme.
We discuss a laser cooling scheme for trapped atoms or ions which is based on double electromagnetically induced transparency (EIT) and makes use of a four-level atom in tripod configuration. The additional fourth atomic state is coupled by a strong coupling laser field to the usual three-level setup of single-EIT cooling. This effectively allows to create two EIT structures in the absorption spectrum of the system to be cooled, which may be controlled by the coupling laser field parameters to cancel both the carrier- and the blue-sideband excitations. In leading order of the Lamb-Dicke expansion, this suppresses all heating processes. As a consequence, the double-EIT scheme can be used to lower the cooling limit by almost two powers of the Lamb-Dicke parameter as compared to single-EIT cooling.
We study the dynamics of a Λ-shaped trapped atom that is driven by two lasers to conditions approaching electromagnetically induced transparency. Solving numerically the full quantum master equation describing the coupled system of electronic and vibrational degrees of freedom, we investigate the dynamics of the entanglement and the quantum mutual information during the cooling process. Far from the Lamb–Dicke limit, an intricate behaviour of these quantities is found, as well as the emergence of a mixed entangled non-equilibrium stationary state.